A moving particle has position

Slides:

Advertisements

Similar presentations

The Calculus of Parametric Equations

Advertisements

When you see… Find the zeros You think…. To find the zeros...

Parametric Equations t x y

Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x

Meanings of the Derivatives. I. The Derivative at the Point as the Slope of the Tangent to the Graph of the Function at the Point.

 Find an equation of the tangent line to the curve at the point (2, 1).

Chapter 10 Review. Directions Take out a sheet of paper to answer all of the review problems. This will be collected for participation points! The “Last.

BC Content Review. When you see… Find the limit of a rational function You think… (If the limit of the top and bottom are BOTH 0 or infinity…)

Normal-Tangential coordinates

Chapter 14 Section 14.3 Curves. x y z To get the equation of the line we need to know two things, a direction vector d and a point on the line P. To find.

Polar Graphs and Calculus

A PREVIEW OF CALCULUS SECTION 2.1. WHAT IS CALCULUS? The mathematics of change.  Velocities  Accelerations  Tangent lines  Slopes  Areas  Volumes.

Section 12.6 – Area and Arclength in Polar Coordinates

Section 8.3 Area and Arc Length in Polar Coordinates.

Section 11.4 Areas and Lengths in Polar Coordinates.

Polar Coordinates.

10.3 Polar Coordinates. Converting Polar to Rectangular Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin.

Calculus with Polar Coordinates Ex. Find all points of intersection of r = 1 – 2cos θ and r = 1.

Warm Up Calculator Active The curve given can be described by the equation r = θ + sin(2θ) for 0 < θ < π, where r is measured in meters and θ is measured.

10.3 Vector Valued Functions Greg Kelly, Hanford High School, Richland, Washington.

Interpreting Motion Graphs {Forces and Motion. Distance vs Time Graphs The motion of an object is defined by its change of position over a period of time.

2006 AP Test. graph of and the line 1. Let R be the shaded region bounded by the as shown at the right. a. Find the area of R.

AP Physics B I.E Circular Motion and Rotation. I.E.1 Uniform Circular Motion.

APPLICATION OF DIFFERENTIATION AND INTEGRATION

1 Chapter 6: Motion in a Plane. 2 Position and Velocity in 2-D Displacement Velocity Average velocity Instantaneous velocity Instantaneous acceleration.

Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson.

Velocity-Time Graphs and Acceleration. What does a v-t graph look like? Time is marked on the horizontal axis and velocity is on the vertical. Graphs.

Cutnell/Johnson Physics 8th edition Reading Quiz Questions

10.3 day 2 Calculus of Polar Curves Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Lady Bird Johnson Grove, Redwood National.

Any vector can be written as a linear combination of two standard unit vectors. The vector v is a linear combination of the vectors i and j. The scalar.

AP Calculus BC Wednesday, 18 November 2015 OBJECTIVE TSW review for the test covering sec – 11.4 and vectors. ASSIGNMENTS DUE FRIDAY –Sec. 11.4:

10. 4 Polar Coordinates and Polar Graphs 10

V ECTORS AND C ALCULUS Section 11-B. Vectors and Derivatives If a smooth curve C is given by the equation Then the slope of C at the point (x, y) is given.

Graphing Polar Graphs Calc AB- Section10.6A. Symmetry Tests for Polar Graphs 1.Symmetry about the x -axis: If the point lies on the graph, the point ________.

How big is my heart??? (Find the area of the enclosed region) WARM UP - Calculator active.

Speeding Up and Slowing Down? Acceleration.

Motion Quiz. 1. The slope of a position (distance) vs time graph equals what quantity of the motion?

Consider the curve defined by Find Write an equation of each horizontal tangent line to the curve The line through the origin with slope –1 is tangent.

FRQ Review. Test Review Retakes by Wed FRQs 7(AB) or 10(BC) types 6 questions per year 9 points each Questions 1 and 2 – calculator Questions 3-6 – non.

9.3: Calculus with Parametric Equations When a curve is defined parametrically, it is still necessary to find slopes of tangents, concavity, area, and.

Parametric equations Parametric equation: x and y expressed in terms of a parameter t, for example, A curve can be described by parametric equations x=x(t),

Calculus with Parametric Curves

Vector-Valued Functions and Motion in Space

1.) Set up the integral to find the area of the shaded region

10.6: The Calculus of Polar Curves

Try graphing this on the TI-89.

Find the area of the region enclosed by one loop of the curve. {image}

10.3 Vector Valued Functions

Motion Along a Line: Vectors

Polar Coordinates and Polar Graphs

Area Between Polar Curves

Parametric Equations and Calculus

Find the velocity of a particle with the given position function

Section 7.1 Day 1-2 Area of a Region Between Two Curves

2.7/2.8 Tangent Lines & Derivatives

CALCULUS BC EXTRA TOPICS

Calculus BC AP/Dual, Revised ©2018

Section 5.3: Finding the Total Area

Warm Up Draw the graph and identify the axis of rotation that

The integral represents the area between the curve and the x-axis.

Review 6.1, 6.2, 6.4.

Motion in a Circle.

Motion Section 3 Acceleration

12.5: Vector PVA.

Warm Up Draw the graph and identify the axis of rotation that

8.3 – Model Periodic Behavior

Chapter 11 Section 11.1 Parametric, Vector, and Polar Functions

10.3 Vector Valued Functions

Plane Curves and Parametric Equations

Presentation transcript:

A moving particle has position at time t. The position of the particle at time t=1 is (2,6). And the velocity vector at any time t>0 is given by a) Find the acceleration vector at time t=3 b) Find the position of the particle at time t=3 c) For what time does the line tangent to the path of the particle at have slope 8? d)The particle approaches a line as Find the slope of this line. Show the work that leads to this conclusion. Bc4 2000

Sketch the polar graph of Find the maximum and minimum values of r Find all points on the curve that have a horizontal tangent

Sketch of the graph of Find to all vertical tangents Find all horizontal tangents

Write the rectangular equation of Find that the slope of the tangent to the curve when Find the area enclosed by the curve Find all points where the curve intersects with the graph of

Find the area inside and outside

Sketch and find the length of from

BC 4 AP 1993 noncalculator Consider the polar curve (a) Sketch the curve (b) Find the area of the region inside the curve (C) Find the slope of the curve at the point where

A roller coaster track has an inverted loop as a portion of its course A roller coaster track has an inverted loop as a portion of its course. The position of the car on the track (in feet) at time t seconds, is given by When is the car at the top of the loop and what is its speed at that time?

At the point where t=2 Determine

Sketch the graph determined by the vector function Find the length of the astroid you just graphed. Find the surface area of the region formed when the curve is rotated about the y-axis

Sketch one arc of the cycloid with equations Find the area under one arc of the cycloid Find the length of one arc of the cycloid Find the surface area of the region determined by rotating one arc of the cycloid about the x-axis Find the volume of the region determined by rotating one arc of the cycloid about the x-axis

Determine the surface area of the region formed by rotating Around the y-axis